Mappings of Butson-type Hadamard matrices

Patric R.J. Östergård; William T. Paavola

doi:10.1016/j.disc.2018.05.012

Mappings of Butson-type Hadamard matrices

Patric R.J. Östergård^*, William T. Paavola

^*Corresponding author for this work

Information Theory

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Abstract

A BH(q,n) Butson-type Hadamard matrix H is an n×n matrix over the complex qth roots of unity that fulfils HH^∗=nI_n. It is well known that a BH(4,n) matrix can be used to construct a BH(2,2n) matrix, that is, a real Hadamard matrix. This method is here generalised to construct a BH(q,pn) matrix from a BH(pq,n) matrix, where q has at most two distinct prime divisors, one of them being p. Moreover, an algorithm for finding the domain of the mapping from its codomain in the case p=q=2 is developed and used to classify the BH(4,16) matrices from a classification of the BH(2,32) matrices.

Original language	English
Pages (from-to)	2387-2397
Number of pages	11
Journal	Discrete Mathematics
Volume	341
Issue number	9
DOIs	https://doi.org/10.1016/j.disc.2018.05.012
Publication status	Published - 1 Sept 2018
MoE publication type	A1 Journal article-refereed

Keywords

Butson-type Hadamard matrix
Classification
Complex matrix
Mapping
Monomial equivalence

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10.1016/j.disc.2018.05.012

ELEC-ostergard-et-al-mappings_of_bustopaper_20180514Accepted author manuscript, 287 KBLicence: CC BY-NC-ND

Cite this

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title = "Mappings of Butson-type Hadamard matrices",

abstract = "A BH(q,n) Butson-type Hadamard matrix H is an n×n matrix over the complex qth roots of unity that fulfils HH∗=nIn. It is well known that a BH(4,n) matrix can be used to construct a BH(2,2n) matrix, that is, a real Hadamard matrix. This method is here generalised to construct a BH(q,pn) matrix from a BH(pq,n) matrix, where q has at most two distinct prime divisors, one of them being p. Moreover, an algorithm for finding the domain of the mapping from its codomain in the case p=q=2 is developed and used to classify the BH(4,16) matrices from a classification of the BH(2,32) matrices.",

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T1 - Mappings of Butson-type Hadamard matrices

AU - Östergård, Patric R.J.

AU - Paavola, William T.

PY - 2018/9/1

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N2 - A BH(q,n) Butson-type Hadamard matrix H is an n×n matrix over the complex qth roots of unity that fulfils HH∗=nIn. It is well known that a BH(4,n) matrix can be used to construct a BH(2,2n) matrix, that is, a real Hadamard matrix. This method is here generalised to construct a BH(q,pn) matrix from a BH(pq,n) matrix, where q has at most two distinct prime divisors, one of them being p. Moreover, an algorithm for finding the domain of the mapping from its codomain in the case p=q=2 is developed and used to classify the BH(4,16) matrices from a classification of the BH(2,32) matrices.

AB - A BH(q,n) Butson-type Hadamard matrix H is an n×n matrix over the complex qth roots of unity that fulfils HH∗=nIn. It is well known that a BH(4,n) matrix can be used to construct a BH(2,2n) matrix, that is, a real Hadamard matrix. This method is here generalised to construct a BH(q,pn) matrix from a BH(pq,n) matrix, where q has at most two distinct prime divisors, one of them being p. Moreover, an algorithm for finding the domain of the mapping from its codomain in the case p=q=2 is developed and used to classify the BH(4,16) matrices from a classification of the BH(2,32) matrices.

KW - Butson-type Hadamard matrix

KW - Classification

KW - Complex matrix

KW - Mapping

KW - Monomial equivalence

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U2 - 10.1016/j.disc.2018.05.012

DO - 10.1016/j.disc.2018.05.012

M3 - Article

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VL - 341

SP - 2387

EP - 2397

JO - Discrete Mathematics

JF - Discrete Mathematics

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ER -

Mappings of Butson-type Hadamard matrices

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Construction and Classification of Discrete Mathematic Structures

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Mappings of Butson-type Hadamard matrices

Abstract

Keywords

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Construction and Classification of Discrete Mathematic Structures

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